spherical shell

**heathrowjohnny** · April 3rd 2008, 07:44 PM

Find the electric field inside a sphere which carries a charge density proportional to the distance from the origin, $\rho = kr$ , for some constant $k$ .

So $\oint \bold{E} \cdot d \bold{a} = \frac{Q_{\text{enc}}}{\epsilon_{0}}$ .

And $E \cdot 4 \pi r^{2} = \frac{1}{\epsilon_{0}} \int \rho \ d \tau$ .

So $E \cdot 4 \pi r^{2} = \frac{k}{\epsilon_{0}} \int_{0}^{2 \pi} \int_{-\pi/2}^{\pi/2} \int_{0}^{r} r'^{3} \sin \theta \ d r' d \theta \ d \phi$ . Then solve for $\bold{E}$ .

Is this correct?

**mr fantastic** · April 3rd 2008, 09:14 PM

Originally Posted by heathrowjohnny

Find the electric field inside a sphere which carries a charge density proportional to the distance from the origin, $\rho = kr$ , for some constant $k$ .

So $\oint \bold{E} \cdot d \bold{a} = \frac{Q_{\text{enc}}}{\epsilon_{0}}$ .

And $E \cdot 4 \pi r^{2} = \frac{1}{\epsilon_{0}} \int \rho \ d \tau$ .

So $E \cdot 4 \pi r^{2} = \frac{k}{\epsilon_{0}} \int_{0}^{2 \pi} \int_{-\pi/2}^{\pi/2} \int_{0}^{r} r'^{3} \sin \theta \ d r' d \theta \ d \phi$ . Then solve for $\bold{E}$ .

Is this correct?

Looks fine. (Although the symbols for azimuthal and polar angles in standard usage are the other way around. But your integral limits and order of integration is consistent with this so there's no problem).

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